436 XV NORMED ALGEBRAS AND SPECTRAL THEORY
for n 2> 0, which are all strictly positive, show that P0(£) == 1 and that
(3) - '^ l
C2 •*• Cn + 1
Cn *•• C2n-i
1
for all n > 1. Also we have
(4)
(e) Conversely, for every (bounded or unbounded) positive measure ^ on R, show
that the operator MM (which is in general unbounded) is self-adjoint and simple.
2. Conversely, let / be a Jacobi matrix (formula (1) of Problem 1) in which the an are
real and the bn are real and >0, but are not restricted in any other way. Let (en)n^o
be the canonical Hilbert basis of the Hilbert space /£ (6.5). Define a hermitian
operator H, with domain the subspace G of /£ generated by the (finite) linear com-
binations of the en, by the conditions
H- en = bn-i en-i + anen + bnen^ (n ^0)
(with e~i = 0, £_! = 0). By abuse of notation, let H also denote the closure of the
hermitian operator H.
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(a) Let £ be a complex number. If y = £ yn en is an eigenvector of H* corresponding
to the eigenvalue £, show that yn = Pn(QyG, where Pn is a polynomial of degree /z,
determined recursively by the formulas PO(£) = 1 and
for all n 2> 0. Deduce that the defect of His (1, 1) if, for some £ e C such that ./£ ^ 0,
we have
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~~ " ; < + oo:
in which case this inequality is true for all £ e C such that ./£ =£ 0. In the contrary
case, the defect of H is (0,0), in other words H is self-adjoint (having regard to the
abuse of language mentioned above). In every case there exist self-adjoint extensions
ofH.
(b) Let F be the subspace of ^C(R) consisting of polynomials with complex coeffi-
cients. Show that there exists a unique linear form ap on F such that
ap(Pn Pm) = $mn (Kronecker index).
(If a polynomial is of the form RS, where R = ]T ut Pt and S = ]T vt PI , then we must
have aF(RS) =2 «* ty, which shows that aF is unique. For the existence, it is enough
to show that, if we write the product U(jc) = (x — z0)R(jc)S(x) in the two forms
((* — z0)R(x))S(x) and R(#)((# — z0)S(x)), where z0 is any complex number, then the
calculation of aF(U) from each of these two products must give the same value.
Reduce to the case where each of R, S is one of the polynomials Pn, and use the
relation (1).) (15.12.8),